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Zero term algebra
Zero term algebra are any algebraic structures where zero terms exists. These algebras (except unary zero term algebras) are often nonassociative due to the controlling features of zero terms on the addition structure. For binary zero term algebras, zero terms form a set of idempotents which is equipped with a one-sided partial ordering defined by their left or right absorption properties with respect to some elements (the D-partial ordering) in the addition structure. Such ordering is often generated by the type of additive identity present. This ordering result in heavy restriction on the form of the addition structure and thus interesting binary zero term algebra are often nonassociative. In contrast, little is known about the underlying theorems that controls the multiplication structure and criteria for nonassociativity. Often the most time consuming step of constructing a binary zero term algebra is to determine the unspecified zero terms to ensure the algebra is self consistent. After this is done, the addition structure can be easily determined using the dominance properties of the zero terms. For nonbinary zero term algebra, the results are well established, including notions of pseudoinverse, possible geometric constructs and embedding of semirings. Zero reciprocal algebra Zero reciprocal algebra contains a unique element called a zero reciprocal which absorbs all elements except the additive identity. Riemann sphere, real projective line, extended nonnegative real number line and any algebra that contains a unique point at infinity are examples, where the point at infinity is the zero reciprocal. Division by zero algebra main article: Division by zero algebra Division by zero algebra is a zero term algebra where the multiplicative identity is a zero term (hence multiplicative inverse of zero elements exists). Most division by zero algebra discovered so far fall into two classes, which their only difference is whether the multiplicative or the addition structure is isomorphic to or contains a one-sided null semigroup. These are the only known associative division by zero algebras. Nonassociative division by zero algebra without one-sided null semigroups remains to be discovered. Divisible zero divisor algebra main article: Divisible zero divisor algebra Divisible zero divisor algebra is a zero term algebra (often a near-semiring with the annihilation axiom discarded to allow existence of zero terms) where zero divisors can be "inverted" via zero terms. For example, suppose a map is defined by "multiplication by a" and its inverse map by "multiplication by b", then ca=0 and 0b=c. Unit zero divisor algebra Unit zero divisor algebra is where a zero divisor has a multiplicative inverse. There are a few known nonassociative examples with some interesting properties such as the existence of one-sided additive inverse and one-sided absorbers thus preventing the zero terms from collapsing to zero, multiplicative absorbers with nilpotents result in invertible maps between zero and the nilpotents (despite not actual inverses). Unary zero term algebra Unary zero term algebras are algebraic structures that contains at least one unary operator that becomes the source of zero terms in that algebra. Wheels One example is Wheels, where division is unary and always defined. Any field can be extended into a wheel. Zero terms are an important feature of Wheels. Unlike binary zero term algebra, the unary division and altered distributive laws for zero terms prevent the zero terms to possess a D-partial order, thus Wheels do not impose much control on their addition structure. No zero terms in wheels except the trivial wheel is equal to the multiplicative identity, however. Other examples Other examples including meadows , where there is a unary operators ~ used to define a pseudoinverse for all elements, and array numbers, where the unary operator is the absence bar, and the additive identity is replaced by an array structure involving the penrose array. Category:Division by zero